How this calculation works
Initialize the level to the first actual. For each later period, score the prior level as its one-step forecast, then update the level using the newly observed actual. The last updated level is the next-period forecast.
L₁ = y₁; forecast yₜ = Lₜ₋₁; Lₜ = αyₜ + (1−α)Lₜ₋₁A worked example
For 10,20,30 with α = 0.5, initialize at 10. Forecast 10 for the second period and update to 15; forecast 15 for the third and update to 22.5. The next forecast is 22.5.
Good to know
At least two equally spaced chronological observations. A fixed entered α, including the explicit boundary cases 0 and 1. No automatic fitting, trend, seasonal correction or uncertainty band. The initializing row is not an error observation.
Reference: NIST: single exponential smoothing (opens in a new tab) ↗
Forecast first and update second
Initialize the level to the first observation, 10. With α = 0.5, the forecast for the second observation is still 10. Only after observing 20 do we update the level to 0.5×20 + 0.5×10 = 15.
The third-period forecast is 15. Observing 30 updates the level to 22.5, which becomes the next-period forecast. Comparing the updated level with the same actual would use information from the period being scored, so the table explicitly shows the prior forecast and the updated level in different columns.
The moving average forecast calculator uses a fixed trailing window; single smoothing recursively retains older information through its preceding level.
The factor changes how new information enters
At α = 1, the level becomes the latest actual after each update. At α = 0, this tool intentionally freezes the first level and later observations do not change it. Intermediate factors blend the latest observation with the preceding level; they do not create a trend term.
For 10,20,30, the two scored errors using predicted minus actual are −10 and −15 at α = 0.5. MAE is 12.5 and RMSE is approximately 12.747549. The initializing first row is not included as an artificial perfect forecast.
A level model needs a sensible comparison
The entered α is not optimized. Trying many factors on the same series and selecting the smallest reported error can overstate future performance. Reserve later observations for evaluation when choosing a real forecasting process and keep the observation interval consistent.
The trailing moving average forecast calculator offers a finite-window baseline. The forecast error calculator can score both methods over the same aligned evaluation periods. Single exponential smoothing alone has no seasonal index, slope, automatically detected change point or prediction interval.
A few questions, answered
Does this select the best smoothing factor?
No. It applies the factor you enter and exposes its one-step errors.
What is the first forecast?
The first scored forecast is for period two, using the first actual as the initialized level.
Can negative observations be used?
Yes, within the supported numeric bounds. Whether a level-only model suits them is a separate modeling decision.